Some conditions on fixed rings

SOME

CONDITIONS

ON

FIXED

RINGS

EDNA E. REITER

Department of Mathematics

University of Cincinnati

Cincinnati, Ohio 45221 U.S.A.

(Received February 12, 1979)

ABSTRACT. This paper deals with a question about the ascending and descending chain conditions on two-sided ideals. Using the ideas of the skew group ring, certain results for two-sided ideals are proved.

KEY WORDS AND PHRASES.

Fixed rings, acending and dcending chain

condon

or two3sided

ideals,

and

skew group

ring.

198o

MATHEMATICS SUBJECT CLASSIFICATION CODES:

i. INTRODUCTION.

Let R be an associative ring with an identity element i, and let G be a group

of automorphisms acting on R such that

IGI,

the order of G, is finite. In this

paper we will prove several relationships between R and the fixed ring RG

{r e

R

rg r for all g e G}. We answer the question raised by Fisher and Osterburg

[7],

that is, we prove that if R has no

IGI

torsion (for r e G,

then we have (Soc R)

[

RGc

qoc R

G.

Thus for a semiprime ring with no

IGI-torsion, Soc R

If

RG Soc R

G.

We also answer a question about the ascending and descending chain conditions on two-sided ideals. It is well known that if

IGI-I

R (i.e., if

IGI

is invertible in R) and if R satisfies the ascending or descending chain condition on left (right) ideals, then RG also satisfies the same condition.

However,

the techniques used for one-sided ideals have not been applied to two-sided ideals. In this paper, we prove the analogous result for two-sided ideals

by using the skew group ring of R and G,’R,G @

Y

Ru Addition in R*G is

componentwise- multiplication is given by the relations (ru)(su

h)

(rsg

)Ugoh

g

extended linearly. The inverse question, posed as follows, remains open: If R is

semiprime and

IGI -I

R, and if RG satisfies the ascending (descending) chain condition on two-sided ideals, then must R satisfy the ascending (descending)

chain condition on two-sided ideals? The

analsous

question is true for one-sided ideals.

For a left and right Noetherian ring, the Artin radical has been defined

as the sum of all Artinian left ideals of the ring. If R is semiprime and

R,

then

A(R)

is defined if and only if

A(R

G)

is defined. In the final section

of this paper, we prove that under these conditions,

A(R)

RG 2. THE SOCLES OF R AND R

G.

First, consider the following theorem relating R and

RGymodules.

The proof is a variation of the proof of

Maschke’s

theorem. Note that all

R,G-modules are certainly R-modules; on the other hand, the left R*G-submodules of R are the G-invariant left ideals of R

(I

R

such that Ig c I

J

g G).

THEOREM 2.1 Let R be a ring and G a finite group of automorphisms of R,

and let @ N /_M /

N’

/ _{0 be a short} exact sequence of left (right)

splits over R, it also splits over R,G.

PROOF. See Fisher-Osterburg

[7,

Theorem

1.3].

They assume

IGI

-I

R, but the proof can be carried out easily without this as long as for any

m

M, N,

there exist

m’

and

n’

in M and N respectively such that

IGlm’

m and

IGln’

n.

From this, we obtain:

COROLLARY 2.2 Let M be a left (or right) R’G-module with

IGI

a

blJection

on

M. If M is semisimple as a left

(or

right) R-module, then M is R,G-semisimple.

PROOF. Immediate.

Now we can apply this to solve the socle question.

THEOREM 2.3 Let R be a ring with no

IGl-torsion,

and let Soc R denote the left (or right) socle of R. Then Soc R

R

G

c

Soc R

G.

PROOF. We give the proof for the left socle only. Note that Soc R is a G-inarlant left ideal of R; thus it is a left R,G-module. The simple components

=

m.

ft

s

of R. Th

=no

b =h=

y

GI

o

GI

.

a

biJection

on Soc R. We can apply Corollary 2.2: Soc R is R,G-semisimple, i.e.,

n

Soc R

=i--

@K

i’ where each Ki is a minimal G-invariant left ideal of R.

We now claim that each K

i

I’1

R

G

is a minimal left ideal of R

G.

For, suppose that X is a left ideal of R

c

such that 0

X_c

K.I

RG"

Then 0

# RX__C

R(K

i

R

G)

___c

Ki;

this implies RX K

i. However, the fact that

GIRX

RX forces

RX

R

G

X, and the claim is proved. n

Now,

(soc

R) RG

(i!%[

@

Ki)

/

RG"

Each K

i

I

R

G

is a minimal of RG

n n

an

.

+/-n f.+/-hd if

.

=a. show ha

(

(R)

Kl

C

e

(K

C

..

i=

xi"

Thus,

xig-

xi

j__

xj

g-xj

O,

i= l,...n, since the sum is

direct and each

i

is G-invariant. These equations show that each x

i

E

R

G.

We note the following corollary.

COROLLARY 2.4 Let R be a semiprime ring with no

Gl-torsion.

Then (Soc R)

G=

Soc R

G.

PROOF. By a theorem of Fisher and Osterburg

[7],

if R is semlprlme with no

Gl-torsion,

then Soc RG

c__

Soc R.

3. CHAIN CONDITIONS ON TWO-SIDED IDEALS.

To deal with two-slded ideals in R and R

G,

we again consider the skew group

ring. In particular, we will use the following lemma.

LEMMA 3.1 Let R be any finite group. If R has the ascending (descending)

chain condition on two-slded ideals, then R,G also satlfles the ascending (descending) chain condition on two-slded ideals.

PROOF. The proof follows the method of the Hilbert Basis Theorem. For G

{gl

gn

},

I

R’G,

we define I

k {r

R:

for some

rl,...rk_

I R,

rlUg

i

+

+

r

k iu

gk-i

+

ru

gk

I} for k i ,n. For a chain of ideals be the kth associated ideal of

I.

as defined above.

I_I,

12

lj

let

lj,k

3

The n chains of ideals I

I,

12,,...

lj,,...,

i, n must each terminate; this forces the chain I. in R,G to terminate.

It may be mentioned that the above lemma is true for any ring T and over-ring S such that there exists a finite subset of S,

{Sl,

Sn

} such that

n

S Ts and s.T Ts..

i=l i z

It is well known (and easily proved) that for any ring S that has the ascendiug (descending) chain condition on two-sided ideals, any subring of the

2

form eSe where e e must also satisfy the same condition. We use this fact, together with Lemma 3.1, to prove the following theorem.

THEOREM 3.2 If

IGI

-I R and R satlfies the ascending (descending) chain conditions on two-sided ideals, then RG satisfies the ascending (descending)

chain condition on two-sided ideals. If

IGI

-I R and R has a composition

PROOF. Consider the element e

(I/IGI)

7

u.

6R,G. It is easy to show G

2

that e e Also, we claim that RG

e(R,G)e.

To prove this, use the ring isomorphism r /

(rUgl)e

where

gl

is the identity of G to obtain RG

RGe

e(R*G)

e. Then, since R,G has the ascending (descending) chain condition on two-sided

ideals, so does R

G.

In the preceeding Theorem 3.2, we may weaken the assumption that R has no

GI

-I

R to the assumption that R has no

Gl-torsion

for the cases of the descending chain condition or the composition series property. This is true

since

IGI

is in the center of R and

RIGI

k,

k 1,2, is a descending chain

of two-sided ideals which must terminate, forcing

IGI

to be invertible in R. We state this as the following corollary.

COROLLARY 3.3. If a ring R has no

IGl-torsion,

then if R satisfies the descending chain condition (has a composition series of two-sided ideals), RG

must also satisfy the descending chain condition (have a composition series of

two-sided ideals).

PROOF. Evident from Theorem 3.2 and the paragraph above.

We conclude this section by noting an example due to Chuang and Lee

[3].

This is a commutative Noetherian ring R with involution, such that R has no 2-torsion and the subring generated by the symmetric elements is not Noetherian.

In a commutative ring, an involution is an automorphism of order 2, and the subring generated by the symmetric elements is the fixed ring. This example,

then, shows that for the ascending chain condition, the hypotheses of Theorem 3o 2 cannot be weakened to assume merely no

4. THE ARTIN RADICALS OF R AND R

G.

In a left and right Noetherian ring R, the Artin radical

A(R)

is defined as the sum of all right ideals of R that are Artinian as right R-modules. This radical turns out to be a two-sided ideal of R; it is also equal to the sum

It is easy to prove that if R is left and right Noetherlan and if

’IGI

-I

R,

then RG is left and right Noetherlan. Also, Farkas and Snlder [5] have proved

that if R is semlprlme with no

Gl-torslon

and if RG is left (right) Noetherlan,

then R is left (right) Noetherlan. Thus, for R semlprlme with

IGI

-I R, A(R) is defined if and only if A(R

G)

is defined. In this section we prove that under these conditions,

A(R)i

RG

A(RG).

To prove the first inclusion, we need several preliminary facts. First,

note that if I is any G-invarlant ideal of R, then G (or a homorphlc image of

G) acts on R/I by g(r

+

I) g(r)

+

I. Furthermore, if

IGI

-I

R,

it can be

shown (see Fisher and Osterburg

[7])

that

(R/I)G

R

G/(I

RG).

Also, we require an important theorem due to Bergman and Isaacs [i], which we state below. The trace of an element is defined by tr x

[

x

g’,

the trace

gG

of an ideal L is {tr x: x L}.

THEOREM 4.1 (Bergman-Isaacs) Let G be a finite group acting on a ring R

without G -torsion. Then:

(i) If R is semlprime, then RG is semlprlme.

(ii) If R is semiprime and L is a G-invariant left ideal of R such that tr

L 0, then L 0. Now we can prove:

THEOREM 4.2 If R is semlprlme and

GI

-I 6R, then A(R)

]

RG

__

A(RG).

PROOF. First, we prove the following fact about right annihilators:

[rR(A(R))]G

rRG[A(R)

RG].

Only one inclusion requires proof. Let x

E

rRG

[A(R)

I

RG];

that is, x RG and [A(R)

i

RG]x

O. Now A(R)x

is a G-invariant left ideal of R and tr (A(R)x) 0. The Bergman-Isaacs is applicable so we have A(R)x 0,

i.e.

x

[rR(A(R))]G.

Noetherian ring such that I is left R-Artinian, then

R/rR(1)

is an Artinian ring.

Thus,

R/rR(A(R))

is an Artinian ring. Since

IGI

-I R, it follows easily that the fixed ring of this factor ring is Artinian. Using this together with the facts on the preceedlng page, we have the Artlnlan rings:

Ir.R

)]

G

RG

And, from the first part of the proof, we have

RG/(rRG[A(R)/

R

G]

is Artlnlan. It is true that a right ideal I of a ring S is contained in A(S) if

S/r(1)

is

Artlnlan (see Chatters, Hajarnavls, and Norton

[2],

the proof of Theorem 1.3).

Applying this, we have A(R) RGc

A(RG).

To prove the other inclusion, we need the following theorem. Its proof is variation of a proof by Cohen and Montgomery [4].

-IEORI 4.3 Let R be a semiprime ring, left and right Noetherian with

IG1-1

R. If A A(R

G)

is the Artinian radical of RG then RA is an Artinian left R-module.

PROOF. Since R is left and right Noetherlan, there are left ideals of R,

K maximal in RA. And, since A is artlnlan, we may choose a finite family

(R

G)

of these, k_I, k

n,

to minimize K

A

(K

I/

Kn

)"

We prove first that K 0.

For,

suppose not. Since

G A is Artlnlan, the socle of RG is essential as a left ideal in A. Also, sinc is semiprime,

Soc (R

G)

is a semlslmple Artlnlan ring and has an identity element i

S Now,

ifK0:

2

K Soc RG (Soc

RG)e,

where e e

#

0

Consider

R(I

S e) RA. The inclusion is proper since the sum Re R(IS e) c RA is direct, and equality would force Re 0 and hence K 0. Therefore,

we can choose a left ideal of R,

K0,

so that K

0 is maximal in RA and:

R(I

We now claim that

el

K0,

which will make the ideal RG

.

(K

0 KI

...

K

n)

properly smaller that K, contradicting the choice of K_I,

Kn

and

thus forcing K 0. To prove the claim, suppose that e K

0" Then obviously

1

S

K0.

Note that the sum A A1S A(I iS is direct, and that Soc R G

RG 1

S, so that Soc R G

A (i i

S 0. Since Soc R G

is essential in

A,

we obtain A A1

S.

Thus, RA R1

s

__

K0,

and RA

K0,

a contradiction to the choice

of K

0.

n

Now consider

Kig

L

__

RA. Certainly L.is a G-Invarlant left

i=l g

G

ideal of R and L A c _K 0. Also, L

I

RG

c RA RG A. We now have:

L RG c L A

f’,

RG 0

Since L RG 0, certainly tr (L R

G)

0. By the Bergman-Isaacs theorem,

L 0. Thus, there is a natural R-module embedding of RA in the finite direct sum of simple left R-module,

I

[

RA/(Kig).

Now it is immediate that RA

+/---

gG

has finite length as a left R-module. We close with two corollaries.

COROLLARY 4.4 Let R be left and right Noetherian, semiprime with

GI

-I

R.

Then A(R

G)

c A(R).

PROOF. We have proved that RA(R

G)

is left Artlnian; A(R) is the sum of all Artlnian left ideals. Thus, A(R

G)

c _RA(R

G)

A(R).

COROLLARY 4.5 let R be left and right Noetherian, smiprime with

IGI

--I,

R.

Then A(R

G)

A(R)

R

G.

REFERENCES

i. G.M. Bergman and I.M.

Isaacs,

Rings with fixed-point free group actions,

Proc. London Math.

Soc.

27

(1973),

pp. 69-87.

2. A.W. Chatters, C.R. Hajarnavis, and N.C. Norton, The Artinian Radical of Noetherian Ring, J. Austral Math.

Soc.

23 (Series

A), (1977),

pp. 379-384. 3. C.L. Chuang and P.H. Lee, Noetherian Rings with Involution, Chinese J. Math.,

5__(1977),

no.

I,

pp. 15-19.

4. Miriam Cohen and Susan Montgomery, Semi-simple Artinian Rings of Fixed Points, Canad. Math.

Bull.

18(2)

(1975),

pp. 189-190.

5. Daniel R. Farkas and Robert L. Snider, Noetherian Fixed Rings, Pacific J.

Math.

69

(1977),

pp. 347-353.

6. Raoul Diop, Actions de Groupes, Doctoral Thesis, L’Universite de Poitiers, 1978.

7. Joe W. Fisher and James Osterburg, Some Results on Rings with Finite Group Actions, Ring Theory: proceedin_s of the Ohio Univ. Conf., Mrc] _Dkr,

(1977), pp. 95-111.

8. T.H. Lenagan, Artinian Ideals in Noetherian Rings, Proc. Amer. Math. Soc.,

51 (1975), pp. 499-500.

9. Martin Lorenz, Primitive Ideals in Crossed Products and Rings with Finite